Mind, Body, World- Foundations of Cognitive Science, 2013a

Descartes did not have at his disposal: the physical symbol system. And—seemingly
magically—a physical symbol system is a finite artifact that is capable of an infinite
variety of behaviour.
By the nineteenth century, the notion of language as a finite system that could
be infinitely expressive was well established (Humboldt, 1999, p. 91): “For language
is quite peculiarly confronted by an unending and truly boundless domain,
the essence of all that can be thought. It must therefore make infinite employment
of finite means.” While Humboldt’s theory of language has been argued to presage
many of the key properties of modern generative grammars (Chomsky, 1966),
it failed to provide a specific answer to the foundational question that it raised:
how can a finite system produce the infinite? The answer to that question required
advances in logic and mathematics that came after Humboldt, and which in turn
were later brought to life by digital computers.
While it had been suspected for centuries that all traditional pure mathematics
can be derived from the basic properties of natural numbers, confirmation of
this suspicion was only obtained with advances that occurred in the nineteenth
and twentieth centuries (Russell, 1993). The “arithmetisation” of mathematics
was established in the nineteenth century, in what are called the Dedekind-Peano
axioms (Dedekind, 1901; Peano, 1973). This mathematical theory defines three
primitive notions: 0, number, and successor. It also defines five basic propositions:
0 is a number; the successor of any number is a number; no two numbers have the
same successor; 0 is not the successor of any number; and the principle of mathematical
induction. These basic ideas were sufficient to generate the entire theory
of natural numbers (Russell, 1993).
Of particular interest to us is the procedure that is used in this system to generate
the set of natural numbers. The set begins with 0. The next number is 1, which
can be defined as the successor of 0, as s(0). The next number is 2, which is the
successor of 1, s(1), and is also the successor of the successor of 0, s(s(0)). In other
words, the successor function can be used to create the entire set of natural numbers:
0, s(0), s(s(0)), s(s(s(0))), and so on.
The definition of natural numbers using the successor function is an example
of simple recursion; a function is recursive when it operates by referring to itself.
The expression s(s(0)) is recursive because the first successor function takes as
input another version of itself. Recursion is one method by which a finite system
(such as the Dedekind-Peano axioms) can produce infinite variety, as in the set of
natural numbers.
Recursion is not limited to the abstract world of mathematics, nor is its only
role to generate infinite variety. It can work in the opposite direction, transforming
the large and complex into the small and simple. For instance, recursion can be
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